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# A Recurrence Relation and the root of a quadratic

Let $$\rho$$ be a root of: $$x^2-2x-1=0$$

Define a function $$F(n) \mid n \in \mathbb N$$ by the recurrence relation:

• $$F(0)=0$$

• $$F(1)=1$$

• $$F(n)= 2F(n-1)+F(n-2) \mid n \geq 2$$

Prove that for all integers $$n\geq1$$:

$\rho^n = F(n-1)+F(n)\rho$

Note by Yan Yau Cheng
3 years, 2 months ago

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